There is one other interesting variant of the binary bisector notation called "hardized binary bisector notation", which mimics and simulates the Hardy hierarchy. It grows slightly slower than the regular binary bisector notation.
The hardized binary bisector notation is defined by the modification of rules 1 and 2 as follows:
Rule 1: A{} = A{0} = A.
Rule 2: A{X + 1} = (2^A){X}, where X can be any sequences greater than natural numbers (1, 2, 3, …), such as {1}.
Examples:
5{3}
= (2^5){2}
= (2^2^5){1}
= (2^2^2^5){0}
= 2^2^2^5
= 2^2^32
= 2^4,294,967,296
≈ 3.103 × 10^1,292,913,986
2{{{1}}}
= 2{{2}}
= 2{{1}.2}
= 2{{1}{1}}
= 2{{1} + 2}
= (2^2){{1} + 1}
= (2^2^2){{1} + 0}
= (2^2^2){{1}}
= (2^4){{1}}
= 16{{1}}
= 16{16}
= (2^16){15}
= (2^2^16){14}
= …
= 2^2^2^…^2^2^2^16 (w/ 16 2’s)
= 2^^19
For some specific fundamental sequences based off the Wainer hierarchy, several properties are satisfied. The rationale is exactly the same as in certain properties of the Wainer hierarchy. In primitive sequence notation level:
For all A, B, if there is no C < A which satisfies A + B = C + B, then N{A + B} = N{B}{A}.
For all A, N{{A}} in the hardized binary bisector notation = N{A} in the regular binary bisector notation.
For example, 4{{{1} + 1}{2}{2}{1}} in the hardized binary bisector notation is equal to 4{{1}}{{2}}{{2}}{{{1} + 1}}, and is also equal to 4{1}{2}{2}{{1} + 1} in the regular binary bisector notation as well.